Theorem onexlimgt 44028

Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexlimgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9622	. . . 4 ⊢ ω ∈ On
2		onun2 6475	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
3	1, 2	mpan2 704	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4		onexomgt 44026	. . 3 ⊢ ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
5	3, 4	syl 18	. 2 ⊢ (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7		omcl 8527	. . . . 5 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
8	1, 6, 7	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9		noel 4291	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∪ ω) ∈ ∅
10		oveq2 7427	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11		om0 8508	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → (ω ·o ∅) = ∅)
12	1, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ω ·o ∅) = ∅
13	10, 12	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
14	13	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
15	14	notbid 321	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
16	15	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
17	9, 16	mpbiri 261	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
18	17	pm2.21d 122	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
19	18	ex 418	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
20	19	com23 87	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21	20	3impia 1135	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22		limom 7884	. . . . . . . . 9 ⊢ Lim ω
23	1, 22	pm3.2i 476	. . . . . . . 8 ⊢ (ω ∈ On ∧ Lim ω)
24	6, 23	jctir 530	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25		omlimcl2 44027	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
26	24, 25	sylan 592	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
27	26	ex 418	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28		on0eqel 6490	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
29	6, 28	syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
30	21, 27, 29	mpjaod 874	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31		simp1 1154	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
32	31, 8	jca 521	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33		simp3 1156	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34		ssun1 4131	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ ω)
35	33, 34	jctil 529	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36		ontr2 6413	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
37	32, 35, 36	sylc 66	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38		limeq 6376	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39		eleq2 2854	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (ω ·o 𝑎)))
40	38, 39	anbi12d 644	. . . . 5 ⊢ (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
41	40	rspcev 3583	. . . 4 ⊢ (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
42	8, 30, 37, 41	syl12anc 850	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
43	42	rexlimdv3a 3172	. 2 ⊢ (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥)))
44	5, 43	mpd 16	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  Oncon0 6364  Lim wlim 6365  (class class class)co 7419  ωcom 7868   ·_o comu 8457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-omul 8464

This theorem is used by: (None)